Question

Given 2^p=5^k=10^r , express p in terms of k and r

Answer 1

we are given

`2^p=5^k=10^r`

Let's assume a term

`4^p`

we can write as

`4^p= (2*2)^p`

`4^p= 2^p*2^p`

now, we know that

`2^p=5^k=10^r`

so, we can replace 2^p

`4^p= 5^k *10^r`

now, we can solve for p

we can take ln on both sides

`ln(4^p)= ln(5^k *10^r)`

now, we can use property of log

`p*ln(4)= ln(5^k)+ln(10^r)`

we can simplify it further

`p*ln(4)= kln(5)+rln(10)`

we get

`p=(kln(5)+rln(10))/(ln(4))`................Answer